416 XV NORMED ALGEBRAS AND SPECTRAL THEORY We have seen in (15.12.4) that T(T)and J(T(T*)) are orthogonal supple- ments of each other in E x E. Hence for each x e E there exists a unique y e dom(T) and a unique z e dom(T*) such that (15.12.6.1) (*, 0) - (y, T-y) + (T* - z, -z). Put y = B - x and z = C - x. Clearly B and C are linear operators defined on the whole of E, and we have B(E) a dom(T) and C(E) <= dom(T*). Also, by (15.12.6.1), W2 = \\y\\2 + \\T-y\\2 + Ni2 + nr* • z||2, so that || 5 • jt|| <| || jc|| and || C • x\\ £ \\x\\ . Hence B and C are continuous. The relation (15.12.6.1) is equivalent to x = B-x+T*C-x and Q=-C-x + TB-x, so that C=TB and T(B(E)) c dom(T*), hence J9(E) <= dom(T*T). Conse- quently T* TB is defined on all of E, and we have which shows that B is injective and /+ T*T surjective. For each w e dom(T*T) we have (15.12.6.2) (w + T*T- w \ w) = |M|2 + (T*T- w \ w) - \\w\\2 + \\T • w\\2 because T= T**; this shows that the relation w + T*T- w = 0 implies that w = 0, and hence that /+ T*T is a bijective mapping of dom(T*r) onto E. Also, since T(B) is closed in E x E (15.12.2), the same is true of F(/+ T*T) (15.12.5), and it follows immediately (15.12.5) that T*Tis closed. We next remark that, for all, u, v in E,,we have (B-u\v) = (B-u\B'V+ T*TB • v) T*T)B -u\B-v) = (u\B-v). Hence B is.self-adjoint. Also, replacing w by B - x in (15.12.6.2) we obtain, for each x e E, e E such that